Galois group: 16T13

• Label: 16T13
• Degree: $16$
• Order: $16$
• Cyclic: no
• Abelian: no
• Solvable: yes
• Primitive: no
• $p$-group: yes
• Group: $D_{8}$

Group invariants

Abstract group:		$D_{8}$
Order:		$16=2^{4}$
Cyclic:		no
Abelian:		no
Solvable:		yes
Nilpotency class:		$3$

Group action invariants

Degree $n$:		$16$
Transitive number $t$:		$13$
Parity:		$1$
Primitive:		no
$\card{\Aut(F/K)}$:		$16$
Generators:		$(1,11)(2,12)(3,10)(4,9)(5,8)(6,7)(13,15)(14,16)$, $(1,4,6,8,10,12,14,15)(2,3,5,7,9,11,13,16)$

Low degree resolvents

$\card{(G/N)}$	Galois groups for stem field(s)
$2$:	$C_2$ x 3
$4$:	$C_2^2$
$8$:	$D_{4}$

Resolvents shown for degrees $\leq 47$

Subfields

Degree 2: $C_2$ x 3

Degree 4: $C_2^2$, $D_{4}$ x 2

Degree 8: $D_4$, $D_{8}$ x 2

Low degree siblings

8T6 x 2

Siblings are shown with degree $\leq 47$

A number field with this Galois group has no arithmetically equivalent fields.

Conjugacy classes

Label	Cycle Type	Size	Order	Index	Representative
1A	$1^{16}$	$1$	$1$	$0$	$()$
2A	$2^{8}$	$1$	$2$	$8$	$( 1,10)( 2, 9)( 3,11)( 4,12)( 5,13)( 6,14)( 7,16)( 8,15)$
2B	$2^{8}$	$4$	$2$	$8$	$( 1,13)( 2,14)( 3,12)( 4,11)( 5,10)( 6, 9)( 7, 8)(15,16)$
2C	$2^{8}$	$4$	$2$	$8$	$( 1,11)( 2,12)( 3,10)( 4, 9)( 5, 8)( 6, 7)(13,15)(14,16)$
4A	$4^{4}$	$2$	$4$	$12$	$( 1, 6,10,14)( 2, 5, 9,13)( 3, 7,11,16)( 4, 8,12,15)$
8A1	$8^{2}$	$2$	$8$	$14$	$( 1, 4, 6, 8,10,12,14,15)( 2, 3, 5, 7, 9,11,13,16)$
8A3	$8^{2}$	$2$	$8$	$14$	$( 1,12, 6,15,10, 4,14, 8)( 2,11, 5,16, 9, 3,13, 7)$

Malle's constant $a(G)$:     $1/8$

Character table

		1A	2A	2B	2C	4A	8A1	8A3
	Size	1	1	4	4	2	2	2
	2 P	1A	1A	1A	1A	2A	4A	4A
	Type
16.7.1a	R	$1$	$1$	$1$	$1$	$1$	$1$	$1$
16.7.1b	R	$1$	$1$	$-1$	$-1$	$1$	$1$	$1$
16.7.1c	R	$1$	$1$	$-1$	$1$	$1$	$-1$	$-1$
16.7.1d	R	$1$	$1$	$1$	$-1$	$1$	$-1$	$-1$
16.7.2a	R	$2$	$2$	$0$	$0$	$-2$	$0$	$0$
16.7.2b1	R	$2$	$-2$	$0$	$0$	$0$	$-{\zeta }_8^{-1}-{\zeta }_8$	${\zeta }_8^{-1}+{\zeta }_8$
16.7.2b2	R	$2$	$-2$	$0$	$0$	$0$	${\zeta }_8^{-1}+{\zeta }_8$	$-{\zeta }_8^{-1}-{\zeta }_8$

Regular extensions

Data not computed